EE538 - Final Report Design of Antenna Arrays using Windowstheshybulb.com/assets/Ant_Report.pdf · 2018. 5. 29. · EE538 - Final Report Design of Antenna Arrays using Windows Mahadevan

EE538 - Final Report

Design of Antenna Arrays using Windows

Mahadevan Srinivasan

April 29, 2010

Abstract

The Design of Uniformly-spaced Antenna Arrays has a significant sim-ilarity to the Design of FIR Filters in Signal Processing and is illustratedfirst. This Paper compares the effect of various windows on the Shapeof the Radiation Pattern of an Antenna Array. To illustrate the powerof windows, we will design a sector pattern and a cosine pattern usingvarious windows available namely Blackmann, Hanning, Hamming andthe customizable Kaiser Window. Comparison is made on the RadiationPatterns produced whilst using each of the Windows using the parametersof the generated pattern namely, Beam width and Side Lobe Level. Todemonstrate the generality of the Window Design method, we see thatthere exists a window which gives an Radiation Pattern with all the sidelobe levels having the same value which gives the same results as theChebyshev Design Method for Uniformly Spaced arrays. Also, there is aWindow for Taylor Array Design Method.

1 Introduction

The Design of Uniformly spaced Arrays is extremely similar to the Problemof FIR Filter Design. So, many methods that are used there can be used inAntenna Array Design. One such method is the Window based FIR FilterDesign. We already know that the Fourier Series Method used in AntennaArray Design is also there in FIR Filter Design. We first explain the FourierSeries Method briefly before going into the Window Method.

2 Fourier Series Method

Suppose it is desired to design a Sector Pattern of beamwidth 60◦

f(θ) ={

1 0◦ ≤ θ ≤ 60◦

0 elsewhere(1)

Design of Array means finding the current excitations required on the ArrayElements to give the required pattern. Since the currents and the RadiationPattern are related through a Fourier Series, we can find the currents using thatrelation.

SFA(w) =N∑

m=−Nime

j2πm dλw (2)

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SFA(w) =N∑m=1

imejπ(2m−1)( dλ )w +

N∑m=1

i−mejπ(2m−1)( dλ )w (3)

As we know, the Fourier Series expansion of a periodic signal x(t) results in anInfinite Number of terms.

cn =1T0

∫ T0/2

−T0/2

x(t)e−jn2πf0t dt (4)

x(t) =∞∑

n=−∞cne

jn2πf0t (5)

But, we can have only finite number of elements in the Array. Hence, wetruncate. This could be viewed as applying a Rectangular Window to the actualFourier Series Expansion. For simplicity, we consider only the Odd Elementscase. It could easily be extended to Even elements case.

SFA(w) =∞∑

m=−∞imwme

j2πm dλw (6)

where, wm is the Window Sequence which is defined as follows.

w(m) ={

1 −N ≤ m ≤ N0 elsewhere

(7)

The Effect of using the Window is seen in the obtained approximate RadiationPattern in the form of a Main Lobe Width and Side Lobe Levels. These are thesignatures of the Rectangular Window Function. Intuitively, the side lobes couldbe explained as due to the abrupt truncation of the Fourier Series Coefficients.

The Problem with using this Window is that the ratio of the Main LobePeak to the Side Lobe Peak is always constant and doesn’t depend upon thenumber of elements we use. So, in an application where there is a need for aside lobe level which is significantly lesser than the Main Lobe Peak, we can usethis Method to design the Array.

3 Window based Design

To reduce the side lobe levels in the generated pattern, we could smoothlytaper the current distribution obtained from the Fourier Series Method usinga properly chosen window. There are several windows available in Literaturewith each having some advantages over the other. Here, we will study the effectof using the following windows : Hann, Hamming, Blackman, Kaiser. Thesewindows are defined by the following equations [1].

Hann

w(n) ={

0.5− 0.5 cos(2πn/N), 0 ≤ n ≤ N0, otherwise

(8)

Hamming

w(n) ={

0.54− 0.46 cos(2πn/N), 0 ≤ n ≤ N0, otherwise

(9)

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Figure 1: Fourier Series Method Output

Blackman

w(n) ={

0.42− 0.5 cos(2πn/N) + 0.08 cos(4πn/N), 0 ≤ n ≤ N0, otherwise

(10)

Through a proper choice of the shape and the length of the window, we couldcontrol the resulting Approximate Radiation Pattern. The Windows defined inthe above Equations are fixed and offer very little in terms of control to the de-signer. This is where the Kaiser Window comes in. It gives one more parameter(β) to the Designer which helps in controlling the shape of the window.

The Compromise between the width of the Main Lobe and the area underthe side lobe can be obtained by having a window function that is concentratedaround 0 in frequency domain. This was considered in classic papers by Slepianet al. (1961). Kaiser(1966,1974) came up with a near-optimal window usingmodified first kind Bessel Function of the zeroth order. It is defined as follows

w(n) =

{I0[β(1−[(n−α)/α]2)1/2]

I0(β) , 0 ≤ n ≤ N

0, otherwise(11)

where α = N/2 and I0(.) is the Modified Bessel Function of zeroth order andfirst kind.

In Figure 2, we have plotted the window function for various values of β anda fixed value of M = N + 1. In Figure 3, we have plotted the fourier transformof the window functions. Finally, we have plotted the fourier transforms ofwindows with same β parameter but different N parameter. For a more detailedreference on Windows, refer to the paper by Harris[2].

Clearly, the parameter β decides the Side-Lobe Level. By keeping β fixedand by increasing N, we could achieve a reduced main lobe width without any

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Figure 2: Kaiser Window Spectrum for various values of β with M = 21

Figure 3: Kaiser Window Spectrum for various values of M with β = 6

effect on the Side Lobe Level. Kaiser also came up with a pair of formulasthat allows the designer to estimate the parameters M and β for the requiredspecification of Peak Approximate Error δ. Peak Approximate Error as thename implies is the maximum error between the desired and the approximateresponse. Peak Approximate Error in a way specifies the maximum side lobe

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Window Beamwidth Sidelobe LevelRectangular 68 -20.8Hamming 85.6 -55.22Hanning 81 -44.03

Blackman 116 -72.63

Table 1: Performance of various windows for Sector Pattern Synthesis

level A = −20log10(δ). The Empirical Relation is as follows,

β =

0.1102(A− 8.7), A > 50,0.5842(A− 21)0.4 + 0.07886(A− 21), 21 ≤ A ≤ 50,0.0, A < 21.

(12)

Kaiser also determined that to achieve the required values of A and the ∆ω(called the Transition Width), N must satisfy

N =A− 8

2.285∆ω(13)

Of Particular Importance is the special windows Chebychev and Taylor.The Use of these windows results in Radiation Pattern which are comparableto the results obtained using the Dolph’s Method [3] and Taylor’s Method [4].We say comparable since we don’t obtain the exact equiripple sidelobes sincemultiplying the Fourier Series Method current coefficients with the Window inequivalent to convolving the Fourier Transform of the Window Function withthat of the Approximate Radiation Pattern.

4 Examples

First, we consider the example of a sector pattern as defined in equation (1)using various fixed windows. We have chosen an array having 21 elements tocompare the performances of various windows. Clearly, we see a reduced sidelobe level compared to the rectangular window output. But, the catch is we havean increased Main Lobe Width. Performance for the Sector Pattern Synthesisusing Window Method is summarized in Table 1.

To see how the use of window affects a different Pattern, we next consider theexample of a cosine pattern [5] with Main Lobe Maximum at 0◦ and a half-powerbeamwidth of 30◦. We see the performance while using various fixed windowsis similar to that obtained for a Sector Pattern in terms of the Beam Width.But, the side lobe level performance is better for the Cosine Pattern synthesis.This can explained as due to the smoothness of the Pattern we started within the first place. Performance for the Cosine Pattern Synthesis using WindowMethod is summarized in Table 2.

4.1 Kaiser Window Design of a Sector Pattern

Suppose it is desired to have a Sector Pattern with Beam width of 60 degreeswith a side lobe level of -26 dB and with a Transition Width of 0.2 (in NormalizedFrequency). The Design Procedure is as follows :

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Window Beamwidth Sidelobe LevelRectangular 68.1116 -39.2285Hamming 85.6873 -63.4069Hanning 81.0832 -61.1224

Blackman 116.4233 -94.7030

Table 2: Performance of various windows for Cosine Pattern Synthesis

Figure 4: Sector Pattern Synthesis using various fixed Windows (Array size 21)

1. Compute the required value of β using equation (12) with A = -26 dB.

2. Compute the number of elements using equation (13) with ∆ω = 0.2.

3. Compute the currents using the Fourier Series Method.

4. Compute the Window Coefficients using the relation in (11).

5. Form the required currents by multiplying the Currents obtained usingthe Fourier Series Method and the window function values.

For the given design specifications, we obtained a value of β = 1.5064 andM = 41. The Approximate Radiation Pattern obtained while using these valuesis shown in Figure 5.

The Kaiser Window Design Method provides a procedure to calculate theNumber of Elements required in the array to achieve the specifications. Toillustrate the improvement obtained by using Kaiser Window, we have plottedthe Radiation Pattern of the Array with size 41 while using a RectangularWindow. Clearly, there is a 6 dB improvement.

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Figure 5: Sector Pattern Synthesis using Chebyshev and Taylor Windows (Arraysize 21)

Figure 6: Cosine Pattern Synthesis using various fixed windows (Array size 21)

5 Conclusion

In essence, all the methods that are available for design of Uniformly spacedarray can be viewed as a special case of the Window Design method with a

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Figure 7: Sector Pattern Synthesis using Kaiser Window (Array Size 41)

proper choice of Window. We have demonstrated the generality by showingexamples of Fourier Series Method and Chebyshev Method. Also, we haveoutlined a procedure for Array design using Kaiser Window where we couldspecify the constraints in the required approximate pattern and could come upwith a Window Function and the length of the array required to realize such aPattern.

References

[1] A. V. Oppenheim and R. W. Schafer, Digital Signal Processing. NJ: Prentice-Hall: Englewood Cliffs, 1975.

[2] F. J. Harris, “On the use of windows for harmonic analysis with discretefourier transform,” IEEE, vol. 66, 1978.

[3] C.L.Dolph, “A current distribution for broadside arrays which optimizes therelationship between beamwith and sidelobe level,” IRE, vol. 34, pp. 335–348, 1946.

[4] T. T. Taylor, “Design of line-source antennas for narrow beamwidth and lowsidelobes,” IRE Transactions Antennas Propagation, vol. AP-3, pp. 16–28,1955.

[5] D. H.Werner and A. J.Ferraro, “Cosine pattern synthesis for single andmultiple main beam uniformly spaced linear arrays,” IEEE, vol. 37, no. 11,pp. 1480–1484, 1989.

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